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A188594
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Decimal expansion of (circumradius)/(inradius) of side-golden right triangle.
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4
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2, 6, 5, 6, 8, 7, 5, 7, 5, 7, 3, 3, 7, 5, 2, 1, 5, 4, 9, 4, 8, 9, 7, 3, 2, 1, 2, 2, 3, 8, 4, 0, 9, 3, 0, 2, 9, 7, 2, 3, 6, 6, 0, 2, 5, 1, 5, 7, 4, 6, 5, 9, 0, 7, 5, 6, 5, 5, 0, 2, 6, 7, 4, 7, 8, 9, 2, 6, 9, 2, 1, 0, 7, 0, 6, 6, 4, 4, 7, 9, 0, 8, 9, 3, 4, 5, 0, 4, 0, 6, 5, 0, 2, 2, 9, 4, 3, 8, 5, 5, 1, 2, 0, 7, 0, 6, 9, 3, 7, 2, 2, 9, 5, 4, 2, 5, 5, 5, 3, 2, 7, 4, 5, 2, 6, 3, 0, 3, 8, 1
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OFFSET
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1,1
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COMMENTS
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This ratio is invariant of the size of the side-golden right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(golden ratio)=(1+sqrt(5))/2. This is the unique right triangle matching the continued fraction [1,1,1,...] of r; i.e, under the side-partitioning procedure described in the 2007 reference, there is exactly 1 removable subtriangle at each stage. (This is analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle as a collection of squares.)
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LINKS
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FORMULA
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(circumradius)/(inradius)=abc(a+b+c)/(8*area^2), where area=area(ABC).
Equals (sqrt(5) + phi*sqrt(2 + phi))/2, where phi = A001622 is the golden ratio. - G. C. Greubel, Nov 23 2017
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EXAMPLE
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2.656875757337521549489732...
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MATHEMATICA
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r=(1+5^(1/2))/2; b=1; a=r*b; c=(a^2+b^2)^(1/2);
area = (1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2);
RealDigits[N[a*b*c*(a+b+c)/(8*area^2), 130]][[1]]
RealDigits[(Sqrt[5] + GoldenRatio*Sqrt[2 + GoldenRatio])/(2), 10, 50][[1]] (* G. C. Greubel, Nov 23 2017 *)
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PROG
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(PARI) {phi = (1 + sqrt(5))/2}; (sqrt(5) + phi*sqrt(2 + phi))/2 \\ G. C. Greubel, Nov 23 2017
(Magma) phi := (1+Sqrt(5))/2; [(Sqrt(5) + phi*Sqrt(2 + phi))/2]; // G. C. Greubel, Nov 23 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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